Wash the cabbage, pat dry and shred it. Transfer to a deep bowl, add salt and sugar and press lightly with your hands.
Add the chopped pepper, carrot and garlic.
Pour in the water with vinegar and the vegetable oil, mix well. At this point taste the cabbage and add more seasoning if needed.
Pack the cabbage firmly into the bowl, press down with a plate, place a weight on top and refrigerate overnight.
In the morning, loosen the cabbage with a fork and transfer to a clean container or jar. Store in the fridge.